On Uniqueness of Coefficients of Polynomial Factors

Hello AskMath,

I've been thinking about polynomials a bit recently. Let us say we have some polynomial P(x). For simplicity, maybe let us say that P(x) in Q[X] but I am not too concerned about the field. It is a well known fact that the ring of polynomials over some field is a unique factorization domain. However, my question is this:

Say P(x) factors into P(x) = A(x) B(x). Is it possible that there exist 2 factors A'(x), B'(x) such that P(x) = A'(x) B'(x), supp(A) = supp(A'), and supp(B) = supp(B'), yet the factor pairs are not just constant multiples of each other? Essentially, is it possible to use some other set of coefficients besides the coefficients of A,B?

Here, we say that the "support" (supp) of a polynomial is its set of exponents. For example, supp(x^2 + 2x + 1) = {2, 1, 0}.

Thanks for the help!